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Metamath Proof Explorer

Theorem nfcsbw

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 12-Oct-2016) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

Ref Expression
Hypotheses nfcsbw.1 ⊢ Ⅎ 𝑥 𝐴
nfcsbw.2 ⊢ Ⅎ 𝑥 𝐵
Assertion nfcsbw Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵

Proof

Step Hyp Ref Expression
1 nfcsbw.1 ⊢ Ⅎ 𝑥 𝐴
2 nfcsbw.2 ⊢ Ⅎ 𝑥 𝐵
3 df-csb ⊢ ⦋ 𝐴 / 𝑦 ⦌ 𝐵 = { 𝑧 ∣ [ 𝐴 / 𝑦 ] 𝑧 ∈ 𝐵 }
4 nftru ⊢ Ⅎ 𝑧 ⊤
5 nftru ⊢ Ⅎ 𝑦 ⊤
6 1 a1i ⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
7 2 a1i ⊢ ( ⊤ → Ⅎ 𝑥 𝐵 )
8 7 nfcrd ⊢ ( ⊤ → Ⅎ 𝑥 𝑧 ∈ 𝐵 )
9 5 6 8 nfsbcdw ⊢ ( ⊤ → Ⅎ 𝑥 [ 𝐴 / 𝑦 ] 𝑧 ∈ 𝐵 )
10 4 9 nfabdw ⊢ ( ⊤ → Ⅎ 𝑥 { 𝑧 ∣ [ 𝐴 / 𝑦 ] 𝑧 ∈ 𝐵 } )
11 3 10 nfcxfrd ⊢ ( ⊤ → Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵 )
12 11 mptru ⊢ Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵